How can I make honey drip down a surface endlessly?
Animated UV map:

One way of going about this is to set up an animated UV map. You will have to have a good material for your honey, and then you animate your UV texture mapping so that the texture appears to be dripping down the sphere.

Here I have started with a UV sphere and kept the default unwrap:


I've set up a simple material using only a texture for the base color. The effect will look much better if you also have normal and roughness textures. If you have the node-wrangler addon enabled (included in Blender, just turn it on in prefs), select your image texture and press CTRL + T to add the Texture Coordinate and Mapping nodes. Plug these into every image texture you end up using.



In terms of the actual animation, you're going to be animating the Y-coordinate of the Mapping node. I inserted a single keyframe at frame 0 with Y=0.0, and a single keyframe at frame 60 (or whatever the last frame of your animation is) with Y=1.0. 

The reason you set the first keyframe at frame 0 even though your animation starts playing at frame 1 is so that you don't have two frames with the same position.

Make sure you set the interpolation mode for these keyframes to linear.

Here is a basic demo of the set up: